Reasoning about distance is indispensable for establishing or avoiding contact in manipulation tasks. To this end, we present an online approach for learning implicit representations of signed distance using piecewise polynomial basis functions. Starting from an arbitrary prior shape, our method incrementally constructs a continuous and smooth distance representation from incoming surface points, with analytical access to gradient information. The underlying model does not store training data for prediction, and its performance can be balanced through interpretable hyperparameters such as polynomial degree and number of segments. We assess the accuracy of the incrementally learned model on a set of household objects and compare it to neural network and Gaussian process counterparts. The utility of intermediate results and analytical gradients is further demonstrated in a physical experiment.